In N dimension,
$$\begin{aligned}
\bm f\times\bm g&\rightarrow f_a\times g_b=[f_ag_b]_{ab}\\
\bm f\cdot\bm g\times\bm h&\rightarrow(f_a, g_b, h_c)=[f_ag_bh_c]_{abc}
\end{aligned}$$
Suppose \(\bm f(\bm r)=f_i\hat e_i\), and a constraint \(f_i\dd r_i=0\)
Even if the \(\nabla\times\bm f\neq 0\), we may find some function \(\varphi\) which is nonzero at the nonzero points of \(\bm f\) s.t.
$$\nabla\times(\varphi\bm f)=\nabla\varphi\times\bm f+\varphi\nabla\times\bm f=0$$
Dotted by \(\bm f\), we find
$$\varphi\bm f\cdot\nabla \times\bm f=(\bm f,\nabla\varphi, \bm f)\propto [f_i\pp_j f_k]_{ijk} =0$$
E.g. For a 2D function, \(A_{ijk}\equiv 0\), so there is always a solution.
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